Generalized Connes-Kreimer Hopf algebras on decorated rooted forests by weighted cocycles
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908696946147328 |
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| author | Wang, Fei Guo, Li Zhang, Yi |
| author_facet | Wang, Fei Guo, Li Zhang, Yi |
| contents | The Connes-Kreimer Hopf algebra of rooted trees is an operated Hopf algebra whose coproduct satisfies the classical Hochschild 1-cocycle condition. In this paper, we extend the setting from rooted trees to the space $H_{\rm RT}(X,Ω)$ of $(X,Ω)$-rooted trees, in which internal vertices are decorated by a set $Ω$ and leafs are decorated by $X \cup Ω$. We introduce a new coalgebra structure on $H_{\rm RT}(X,Ω)$ whose coproduct satisfies a weighted Hochschild 1-cocycle condition involving multiple operators, thereby generalizing the classical condition. A combinatorial interpretation of this coproduct is also provided. We then endow $H_{\rm RT}(X,Ω)$ with a Hopf algebra structure. Finally, we define weighted $Ω$-cocycle Hopf algebras, characterized by a Hochschild 1-cocycle condition with weights, and show that $H_{\rm RT}(X,Ω)$ is the free object in the category of $Ω$-cocycle Hopf algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06538 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Connes-Kreimer Hopf algebras on decorated rooted forests by weighted cocycles Wang, Fei Guo, Li Zhang, Yi Quantum Algebra Combinatorics Rings and Algebras 16T30, 17B38, 05E16, 05C05, 16S10 The Connes-Kreimer Hopf algebra of rooted trees is an operated Hopf algebra whose coproduct satisfies the classical Hochschild 1-cocycle condition. In this paper, we extend the setting from rooted trees to the space $H_{\rm RT}(X,Ω)$ of $(X,Ω)$-rooted trees, in which internal vertices are decorated by a set $Ω$ and leafs are decorated by $X \cup Ω$. We introduce a new coalgebra structure on $H_{\rm RT}(X,Ω)$ whose coproduct satisfies a weighted Hochschild 1-cocycle condition involving multiple operators, thereby generalizing the classical condition. A combinatorial interpretation of this coproduct is also provided. We then endow $H_{\rm RT}(X,Ω)$ with a Hopf algebra structure. Finally, we define weighted $Ω$-cocycle Hopf algebras, characterized by a Hochschild 1-cocycle condition with weights, and show that $H_{\rm RT}(X,Ω)$ is the free object in the category of $Ω$-cocycle Hopf algebras. |
| title | Generalized Connes-Kreimer Hopf algebras on decorated rooted forests by weighted cocycles |
| topic | Quantum Algebra Combinatorics Rings and Algebras 16T30, 17B38, 05E16, 05C05, 16S10 |
| url | https://arxiv.org/abs/2512.06538 |